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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E30</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="radix_1.html#R1">is_represented_by</a> 1,<font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:9"/><i><font color="Green">E31</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then </b><i><font color="Green">E33</font></i>:  <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>,1 = 
 <a href="radix_3.html#K9">SD2SDSubDigitS</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="radix_3.html#D8">RADIX_3:def 8</a></i>
<br/>.= 
 <a href="radix_3.html#K8">SD2SDSubDigit</a> <span class="p1">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1,<font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="radix_3.html#D7">RADIX_3:def 7</a></i>
;<br/>
<a NAME="E4:9"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b> <a href="radix_3.html#K7">DigA_SDSub</a> <span class="p1">(<span class="default"><a href="radix_3.html#K10">SD2SDSub</a> <span class="p2">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span></span>)</span>,1 = 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,<span class="p3">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="radix_3.html#D6">RADIX_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,0</span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <span class="p2">(<span class="default"><a href="radix_1.html#K4">DigA</a> <span class="p3">(<span class="default"><a href="radix_1.html#K10">DecSD</a> <font color="Maroon">m</font>,1,<font color="Maroon">k</font></span>)</span>,1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> 0,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="radix_1.html#D3">RADIX_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> 0,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="radix_1.html#T24">RADIX_1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="radix_3.html#K6">SDSub_Add_Data</a> <font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
<b>by </b><i>, <a class="ref" href="radix_3.html#T17">RADIX_3:17</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="radix_3.html#K5">SDSub_Add_Carry</a> <font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="radix_1.html#K1">Radix</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="radix_3.html#D4">RADIX_3:def 4</a></i>
;<br/><br/>


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